Look east from the Kurdish trenches on a dusty ridge outside the northern Iraqi city of Kirkuk and you can see the cause of it all: a rudimentary oil field where water wells are being sunk and sites are being cleared for drilling.

Now look south into the valley below the Kurdish positions and you can see two Iraqi army units poised to make sure that drilling never begins.

Since November, a crisis of oil, money and history has been building in the semiautonomous northern Iraqi region of Kurdistan. Some 30,000 Kurdish soldiers face just as many regular Iraqi army troops,...